Galois stratification over Frobenius fields

Abstract

Let K be an infinite field finitely generated over its prime lield. Denote by G(K) = .‘a(K,/K) the absolute Galois group of K. The set G(K)‘, for e a positive integer, is equipped with the normalized Haar measure, ,U =,uu,, induced from the measure of G(K) that assigns to G(L) the value l/[L: K], if L/K is a finite separable extension. If o = (a, ,..., a,) E G(K)‘, then we denote by g(a) the fixed field of u,,..., ue in 2 (= the algebraic closure of K). Denote also by Y(K) the first-order language of fields enriched with constant symbols for the elements of K. For every sentence 0 of Y(K) we define A,(B) = {a E G(K)=Il?(u) i= 19). Further we denote by T,(K) the theory of all sentences 19 of Y(K) with ,u(A,(O)) = 1. In [ 13, Theorem 7.3 1 the following is shown.